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6. Show that a simple group is solvable iff it is cyclic. a b 7. Show that G = 0 1 c a, b, cE Z3 is a solvable group. (Hint: o(G) = 33) 10 0 8. If p, q are primes, show that groups of order p q, p q are solvable. 9. Give example of a group all of whose proper subgroups are solvable but group itself is not. (Consider A;). 10. If all proper subgroups of a non solvable group G are solvable, show that G = G'. (A group G such that G = G' is called a perfect group). 11. Show that every group of odd order is solvable iff every finite non abelian simple group has even order. 12. Show that direct product of infinitely many solvable groups need not be solvable. 13. Show that a finite p-group is nilpotent. 14. Show that the result proved in theorem 14 for solvable groups does not hold for nilpotent groups. [Hint: Consider S3/A3] 15. Give an example of a solvable group G in which H S G, H # G and N(H) # H. 16. Suppose that in a non abelian simple group, fe; is the only conjugate class whose order is prime power. Show that a group of order p"q" (p, q primes) is a solvable group. 17. Let G be a nilpotent group. Show that every maximal subgroup of G is normal subgroup. Hence deduce that S, is not nilpotent. (Use problem 31) 18. Show that every sylow subgroup of a nilpotent group G is normal in G. (See exercise 17) 19. Show that a nilpotent group is isomorphic to the direct product of its Sylow subgroups. 20. If G is direct product of its Sylow subgroups, show that G is nilpotent. (Use exercise 13 and problem 30)